This document covers the topic of indeterminate forms and L'Hôpital’s rule in calculus. It explains different types of indeterminate limits, provides examples, and discusses how to resolve these limits using L'Hôpital's rule. Additionally, it highlights the importance of identifying when limits are indeterminate and warns against applying the rule to determinate limits.

1. . V63.0121.001: Calculus I3.7: Indeterminate forms and lHôpital’s Rule
. Sec on .
Notes
Sec on 3.7
Indeterminate forms and lHôpital’s
Rule
V63.0121.001: Calculus I
Professor Ma hew Leingang
New York University
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Notes
Announcements
Midterm has been
returned. Please see FAQ
on Blackboard (under
”Exams and Quizzes”)
Quiz 3 this week in
recita on on Sec on 2.6,
2.8, 3.1, 3.2
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Notes
Objectives
Know when a limit is of
indeterminate form:
indeterminate quo ents:
0/0, ∞/∞
indeterminate products:
0×∞
indeterminate
differences: ∞ − ∞
indeterminate powers:
00 , ∞0 , and 1∞
Resolve limits in
indeterminate form
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2. . V63.0121.001: Calculus I3.7: Indeterminate forms and lHôpital’s Rule
. Sec on .
Notes
Recall
Recall the limit laws from Chapter 2.
Limit of a sum is the sum of the limits
Limit of a difference is the difference of the limits
Limit of a product is the product of the limits
Limit of a quo ent is the quo ent of the limits ... whoops! This
is true as long as you don’t try to divide by zero.
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Notes
More about dividing limits
We know dividing by zero is bad.
Most of the me, if an expression’s numerator approaches a
finite nonzero number and denominator approaches zero, the
quo ent has an infinite. For example:
1 cos x
lim+ = +∞ lim− = −∞
x→0 x x→0 x3
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Notes
Why 1/0 ̸= ∞
1
Consider the func on f(x) = 1 .
x sin x
y
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x
Then lim f(x) is of the form 1/0, but the limit does not exist and is
x→∞
not infinite.
Even less predictable: when numerator and denominator both go to
zero.
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3. . V63.0121.001: Calculus I3.7: Indeterminate forms and lHôpital’s Rule
. Sec on .
Notes
Experiments with funny limits
sin2 x
lim =0
x→0 x
x
lim does not exist
x→0 sin2 x
2
sin x .
lim 2)
=1
x→0 sin(x
sin 3x
lim =3
x→0 sin x
0
All of these are of the form , and since we can get different
0
answers in different cases, we say this form is indeterminate.
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Notes
Experiments with funny limits
sin2 x
lim =0
x→0 x
x
lim does not exist
x→0 sin2 x
2
sin x .
lim =1
x→0 sin(x2 )
sin 3x
lim =3
x→0 sin x
0
All of these are of the form , and since we can get different
0
answers in different cases, we say this form is indeterminate.
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Notes
Experiments with funny limits
sin2 x
lim =0
x→0 x
x
lim does not exist
x→0 sin2 x
2
sin x .
lim 2)
=1
x→0 sin(x
sin 3x
lim =3
x→0 sin x
0
All of these are of the form , and since we can get different
0
answers in different cases, we say this form is indeterminate.
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4. . V63.0121.001: Calculus I3.7: Indeterminate forms and lHôpital’s Rule
. Sec on .
Notes
Experiments with funny limits
sin2 x
lim =0
x→0 x
x
lim does not exist
x→0 sin2 x
2
sin x .
lim 2)
=1
x→0 sin(x
sin 3x
lim =3
x→0 sin x
0
All of these are of the form , and since we can get different
0
answers in different cases, we say this form is indeterminate.
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Language Note Notes
It depends on what the meaning of the word “is” is
Be careful with the language here. We
are not saying that the limit in each
0
case “is” , and therefore nonexistent
0
because this expression is undefined.
0
The limit is of the form , which means
0
we cannot evaluate it with our limit
laws.
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Notes
Indeterminate forms are like Tug Of War
. Which side wins depends on which side is stronger.
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5. . V63.0121.001: Calculus I3.7: Indeterminate forms and lHôpital’s Rule
. Sec on .
Notes
Outline
L’Hôpital’s Rule
Rela ve Rates of Growth
Other Indeterminate Limits
Indeterminate Products
Indeterminate Differences
Indeterminate Powers
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Notes
The Linear Case
Ques on
f(x)
If f and g are lines and f(a) = g(a) = 0, what is lim ?
x→a g(x)
Solu on
The func ons f and g can be wri en in the form
f(x) = m1 (x − a) g(x) = m2 (x − a)
So
f(x) m1
=
g(x) m2
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Notes
The Linear Case, Illustrated
y y = g(x)
y = f(x)
a f(x) g(x)
. x
x
f(x) f(x) − f(a) (f(x) − f(a))/(x − a) m1
= = =
g(x) g(x) − g(a) (g(x) − g(a))/(x − a) m2
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6. . V63.0121.001: Calculus I3.7: Indeterminate forms and lHôpital’s Rule
. Sec on .
Notes
What then?
But what if the func ons aren’t linear?
Can we approximate a func on near a point with a linear
func on?
What would be the slope of that linear func on?
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Notes
Theorem of the Day
Theorem (L’Hopital’s Rule)
Suppose f and g are differen able func ons and g′ (x) ̸= 0 near a
(except possibly at a). Suppose that
lim f(x) = 0 and lim g(x) = 0
x→a x→a
or lim f(x) = ±∞ and lim g(x) = ±∞
x→a x→a
Then
f(x) f′ (x)
lim = lim ′ ,
x→a g(x) x→a g (x)
if the limit on the right-hand side is finite, ∞, or −∞.
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Notes
Meet the Mathematician
wanted to be a military man, but
poor eyesight forced him into
math
did some math on his own
(solved the “brachistocrone
problem”)
paid a s pend to Johann
Bernoulli, who proved this Guillaume François Antoine,
theorem and named it a er him! Marquis de L’Hôpital
(French, 1661–1704)
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7. . V63.0121.001: Calculus I3.7: Indeterminate forms and lHôpital’s Rule
. Sec on .
Notes
Revisiting the previous examples
Example
sin2 x H 2 sin x cos x
lim = lim =0
x→0 x x→0 1
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Notes
Revisiting the previous examples
Example
sin2 x H sin x cos x H
2 cos2 x − sin2 x
lim = lim = lim =1
x→0 sin x 2 x→0 (cos x 2 ) (2x)
x→0 cos x2 − 2x2 sin(x2 )
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Notes
Revisiting the previous examples
Example
sin 3x H 3 cos 3x
lim = lim = 3.
x→0 sin x x→0 cos x
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8. . V63.0121.001: Calculus I3.7: Indeterminate forms and lHôpital’s Rule
. Sec on .
Notes
Beware of Red Herrings
Example
Find
x
lim
x→0 cos x
Solu on
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Notes
Outline
L’Hôpital’s Rule
Rela ve Rates of Growth
Other Indeterminate Limits
Indeterminate Products
Indeterminate Differences
Indeterminate Powers
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Limits of Rational Functions Notes
revisited
Example
5x2 + 3x − 1
Find lim if it exists.
x→∞ 3x2 + 7x + 27
Solu on
Using L’Hôpital:
5x2 + 3x − 1 H 10x + 3 H 10 5
lim = lim = lim =
x→∞ 3x2 + 7x + 27 x→∞ 6x + 7 x→∞ 6 3
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9. . V63.0121.001: Calculus I3.7: Indeterminate forms and lHôpital’s Rule
. Sec on .
Limits of Rational Functions Notes
revisited II
Example
5x2 + 3x − 1
Find lim if it exists.
x→∞ 7x + 27
Solu on
Using L’Hôpital:
5x2 + 3x − 1 H 10x + 3
lim = lim =∞
x→∞ 7x + 27 x→∞ 7
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Limits of Rational Functions Notes
revisited III
Example
4x + 7
Find lim if it exists.
x→∞ 3x2 + 7x + 27
Solu on
Using L’Hôpital:
4x + 7 H 4
lim = lim =0
x→∞ 3x2 + 7x + 27 x→∞ 6x + 7
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Notes
Limits of Rational Functions
Fact
Let f(x) and g(x) be polynomials of degree p and q.
f(x)
If p q, then lim =∞
x→∞ g(x)
f(x)
If p q, then lim =0
x→∞ g(x)
f(x)
If p = q, then lim is the ra o of the leading coefficients of
x→∞ g(x)
f and g.
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10. . V63.0121.001: Calculus I3.7: Indeterminate forms and lHôpital’s Rule
. Sec on .
Notes
Exponential vs. geometric growth
Example
ex
Find lim , if it exists.
x→∞ x2
Solu on
We have
ex H ex H ex
lim = lim = lim = ∞.
x→∞ x2 x→∞ 2x x→∞ 2
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Notes
Exponential vs. geometric growth
Example
ex
What about lim ?
x→∞ x3
Answer
Solu on
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Notes
Exponential vs. fractional powers
Example
ex
Find lim √ , if it exists.
x→∞ x
Solu on (without L’Hôpital)
We have for all x 1, x1/2 x1 , so
ex ex
1/2
x x
The right hand side tends to ∞, so the le -hand side must, too.
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11. . V63.0121.001: Calculus I3.7: Indeterminate forms and lHôpital’s Rule
. Sec on .
Notes
Exponential vs. fractional powers
Example
ex
Find lim √ , if it exists.
x→∞ x
Solu on (with L’Hôpital)
ex ex √
lim √ = lim 1 −1/2 = lim 2 xex = ∞
x→∞ x x→∞ x
2
x→∞
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Notes
Exponential vs. any power
Theorem
ex
Let r be any posi ve number. Then lim = ∞.
x→∞ xr
Proof.
If r is a posi ve integer, then apply L’Hôpital’s rule r mes to the frac-
on. You get
ex H H ex
lim = . . . = lim = ∞.
x→∞ xr x→∞ r!
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Notes
Exponential vs. any power
Theorem
ex
Let r be any posi ve number. Then lim = ∞.
x→∞ xr
Proof.
If r is not an integer, let m be the smallest integer greater than r. Then
ex ex
if x 1, xr xm , so r m . The right-hand side tends to ∞ by the
x x
previous step.
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12. . V63.0121.001: Calculus I3.7: Indeterminate forms and lHôpital’s Rule
. Sec on .
Notes
Any exponential vs. any power
Theorem
ax
Let a 1 and r 0. Then lim = ∞.
x→∞ xr
Proof.
If r is a posi ve integer, we have
ax H H (ln a)r ax
lim = . . . = lim = ∞.
x→∞ xr x→∞ r!
If r isn’t an integer, we can compare it as before.
(1.00000001)x
So even lim = ∞!
. x→∞ x100000000
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Notes
Logarithmic versus power growth
Theorem
ln x
Let r be any posi ve number. Then lim = 0.
x→∞ xr
Proof.
One applica on of L’Hôpital’s Rule here suffices:
ln x H 1/x 1
lim = lim r−1 = lim r = 0.
x→∞ xr x→∞ rx x→∞ rx
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Notes
Outline
L’Hôpital’s Rule
Rela ve Rates of Growth
Other Indeterminate Limits
Indeterminate Products
Indeterminate Differences
Indeterminate Powers
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13. . V63.0121.001: Calculus I3.7: Indeterminate forms and lHôpital’s Rule
. Sec on .
Notes
Indeterminate products
Example
√
Find lim+ x ln x
x→0
This limit is of the form 0 · (−∞).
Solu on
Jury-rig the expression to make an indeterminate quo ent. Then
apply L’Hôpital’s Rule:
√ ln x H x−1 √
lim x ln x = lim+ 1 √ = lim+ 1 −3/2 = lim+ −2 x = 0
x→0+ x→0 / x x→0 − 2 x x→0
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Notes
Indeterminate differences
Example
( )
1
lim − cot 2x
x→0+ x
This limit is of the form ∞ − ∞.
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Notes
Indeterminate Differences
Solu on
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14. . V63.0121.001: Calculus I3.7: Indeterminate forms and lHôpital’s Rule
. Sec on .
Notes
Indeterminate powers
Example
Find lim+ (1 − 2x)1/x
x→0
Solu on
Take the logarithm:
( ) ( ) ln(1 − 2x)
ln lim+ (1 − 2x)1/x = lim+ ln (1 − 2x)1/x = lim+
x→0 x→0 x→0 x
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Notes
Indeterminate powers
Example
Find lim+ (1 − 2x)1/x
x→0
Solu on
0
This limit is of the form , so we can use L’Hôpital:
0
−2
ln(1 − 2x) H
lim+ = lim+ 1−2x = −2
x→0 x x→0 1
This is not the answer, it’s the log of the answer! So the answer we
. want is e−2 .
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Notes
Another indeterminate power limit
Example
Find lim (3x)4x
x→0
Solu on
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15. . V63.0121.001: Calculus I3.7: Indeterminate forms and lHôpital’s Rule
. Sec on .
Notes
Summary
Form Method
0
0 L’Hôpital’s rule directly
∞
∞ L’Hôpital’s rule directly
∞
0·∞ jiggle to make 0 or
0 ∞.
∞ − ∞ combine to make an indeterminate product or quo ent
00 take ln to make an indeterminate product
∞0 di o
1∞ di o
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Notes
Final Thoughts
L’Hôpital’s Rule only works on indeterminate quo ents
Luckily, most indeterminate limits can be transformed into
indeterminate quo ents
L’Hôpital’s Rule gives wrong answers for non-indeterminate
limits!
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Notes
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